I understand mathematically what $\limsup A_n$ means, it's the set of outcomes that occur infinitely many times. However, I'm trying to intuit the meaning via a simple example: Let $A_n$ be a sequence of events such that $$A_n = \{\text{"heads" not seen in first $n$ flips}\}$$ What would $\limsup A_n$ be in this case? In this case isn't the event space for $A_n$ simply $\{\text{True}, \text{False} \}$ And so $\limsup A_n = \{\text{True}, \text{False} \}$?
Perhaps this isn't the greatest example, so any other elementary examples to understand the meaning of limsup for events would be greatly appreciated.